1. Field of the Invention
The invention relates especially to a method of fourth-order and higher order self-learned (or blind) separation of sources from reception with N receivers or sensors (N≧2), this method exploiting no a priori information on sources or wavefronts, and the sources being P cyclostationary (deterministic or stochastic, analog or digital sources with linear or non-linear modulation) and statistically independent sources,
It can be applied for example in the field of radio communications, space telecommunications or passive listening to these links in frequencies ranging for example from VLF to EHF.
It can also be applied in fields such as astronomy, biomedicine, radar, speech processing, etc.
2. Description of the Prior Art
The blind separation of sources and, more particularly, independent component analysis (ICA) is currently arousing much interest. Indeed, it can be used in many applications such as telecommunications, speech processing or again biomedicine.
For example, in antenna processing, if signals sent from a certain number of sources are received at an array of receivers and if, for each source, the timing spread of the channels associated with a different receivers is negligible as compared with the timing symbol time, then an instantaneous mixture of the signals sent from the sources is observed on said receivers.
The blind separation of sources is aimed especially at restoring the sources assumed to be statistically independent, and this is done solely on the basis of the observations received by the receivers.
Depending on the application, it is possible to retrieve only the instantaneous mixture, namely the direction vectors of the sources. This is the case for example with goniometry where said mixture carries all the information needed for the angular localization of the sources by itself: the term used then is “blind identification of mixtures”.
For other applications such as transmission, it is necessary to retrieve the signals sent from the sources: the expression used then is separation or again blind or self-learned extraction of sources.
Certain prior art techniques seek to carry out a second-order decorrelation (as can be seen in factor analysis with principal component analysis (PCA).
ICA, for its part, seeks to reduce the statistical dependence of the signals also at the higher orders. Consequently, ICA enables the blind identification of the instantaneous mixture and thereby the extraction of the signals sent from the sources, not more than one of which is assumed to be Gaussian. At present, this is possible only in complying with certain assumptions: the noisy mixture of the sources must be linear and furthermore over-determined (the number of sources P must be smaller than or equal to the number of receivers N).
While Comon was the first to introduce the ICA concept and propose a solution, COM2 in the reference [1] (the different references are brought together at the end of the description) maximized a contrast based on fourth-order cumulants, Cardoso and Souloumiac [2], for their part developed a matrix approach, better known as JADE, and thus created the joint diagonalization algorithm.
Some years later, Hyvarinen et al. presented the FastICA method, initially for real signals [3], and then in complex cases [4]. This method introduces a contrast-optimizing algorithm called the fixed-point algorithm.
Comon has proposed a simple solution, COM1 [5], to contrast optimization presented in [6].
Although these methods perform very well under the assumptions stated here above, they may nevertheless be greatly disturbed by the presence of unknown noise, whether Gaussian or not, that is spatially correlated and inherent in certain applications such as HF radio communications.
Furthermore, as stated further above, the above methods are designed only to process over-determined mixtures of sources. Now in practice, for example in radio communications, it is not rare to have reception from more sources than receivers, especially if the reception bandwidth is great. We then have what are called under-determined mixtures (P>N).
Several algorithms have been developed already in order to process mixtures of this type. Some of them tackle the difficult problem of the extraction of sources [7–8] when the mixture is no longer linearly inverted, while others deal with the indication of the mixture matrix [7] [9–12].
The methods proposed in [9–11] exploit only fourth-order statistics while the method presented in [12] relies on the use of the characteristic second function of the observations, in other words on the use of non-zero cumulants of any order. As for the method used in [7], it relies on the conditional maximization of probability, in this case that of data conditional on the mixture matrix.
While these methods perform well, they have drawbacks in the operational context.
Thus, the method [9] is difficult to implement and does not ensure the identification of the direction vectors of sources of the same kurtosis. The methods [10] and [11] for their part cannot be used to identify the direction vectors of circular sources. The method [10], called S3C2, for its part confines the user in a configuration of three sources and two receivers, ruling out any other scenario. The method [7] authorizes the identification of four speech signals with only two receivers. However the samples observed must be temporally independent and each source must have a sparse density of probability. Finally, the method [12] is applicable only in the case of real sources, which is highly restrictive especially in digital communications. Furthermore, the algorithm depends greatly on the number of sources, and there is nothing today to prove that a poor estimation of this parameter will not impair the performance of the method.